Prerequisites
Almost essential
Firm: Optimisation
THE FIRM: DEMAND AND
SUPPLY
MICROECONOMICS
Principles and Analysis
Frank Cowell
March 2012
Frank Cowell: Firm- Demand & Supply
1
Moving on from the optimum…
 We derive the firm's reactions to changes in its
environment
 These are the response functions
• We will examine three types of them
• Responses to different types of market events
 In effect we treat the firm as a black box
the firm
March 2012
Frank Cowell: Firm- Demand & Supply
2
The firm as a “black box”
 Behaviour can be predicted by necessary and sufficient




conditions for optimum
The FOC can be solved to yield behavioural response
functions
Their properties derive from the solution function
We need the solution function’s properties…
…again and again
March 2012
Frank Cowell: Firm- Demand & Supply
3
Overview…
Firm: Comparative
Statics
Conditional
Input Demand
Response
function for stage
1 optimisation
Output
Supply
Ordinary
Input Demand
Short-run
problem
March 2012
Frank Cowell: Firm- Demand & Supply
4
The first response function

Review the cost-minimisation
problem and its solution

Choose z to minimise
The “stage 1” problem
m
S wi zi subject to q  f (z), z ≥ 0
i=1

The firm’s cost function:
The solution function
C(w, q) := min S wizi
{f(z)  q}

Cost-minimising value for each input:
zi* = Hi(w, q), i=1,2,…,m
could be a welldefined function or a
correspondence
March 2012
Specified
vector of output level
input prices
 Hi is the conditional input
demand function
Demand for input i, conditional
on given output level q
A graphical
approach
Frank Cowell: Firm- Demand & Supply
5
Mapping into (z1,w1)-space
Conventional case of Z
Start with any value of w1 ( the
slope of the tangent to Z)
Repeat for a lower value of w1
…and again to get…
z2
w1
…the conditional demand curve
 Constraint set is
convex, with smooth
boundary


 Response function is
a continuous map:

z1
H1(w,q)
z1
Now try a
different case
March 2012
Frank Cowell: Firm- Demand & Supply
6
Another map into (z1,w1)-space
Now take case of nonconvex Z
Start with a high value of w1
Repeat for a very low value of w1
Points “nearby” work the same
way
z2
 But what happens in between?
w1
A demand correspondence
 Constraint set is nonconvex


 Response is discontinuous
map: jumps in z*
Multiple inputs
at this price




z1
Map is multivalued at the
discontinuity
z1
no price yields
a solution here
March 2012
Frank Cowell: Firm- Demand & Supply
7
Conditional input demand function
 Assume that single-valued input-demand functions exist
 How are they related to the cost function C?
 What are their properties?
 How are their properties related to those of C?
• tip if you’re not sure about the cost function…
• …check the presentation “Firm Optimisation”
• …revise the five main properties of the function C
March 2012
Frank Cowell: Firm- Demand & Supply
8
Use the cost function

Recall this relationship?
…yes, it's Shephard's lemma
Ci(w, q) = zi*
The slope:
 C(w, q)
————
 wi

Optimal demand
for input i
So we have:
Ci(w, q) =
Hi(w,
q)
Link between conditional input
demand and cost functions
conditional input
demand function
 Differentiate this with respect to wj
Cij(w, q) = Hji(w, q)
Slope of input demand function
Second
derivative
March 2012
Two simple
results:
Frank Cowell: Firm- Demand & Supply
9
Simple result 1
 Use a standard property
second derivatives of a function
“commute”
 So in this case
The order of differentiation is irrelevant
 2()
2()
——— = ———
 wi wj
wj wi
Cij(w, q) = Cji(w,q)
 Therefore we have:
Hji(w, q) = Hij(w, q)
March 2012
The effect of the price of input i on
conditional demand for input j equals
the effect of the price of input j on
conditional demand for input i
Frank Cowell: Firm- Demand & Supply
10
Simple result 2


Cij(w, q) = Hji(w, q)
Slope of conditional input
demand function derived from
second derivative of cost function
We can get the special case:
We've just put j = i
Use the standard relationship:
Cii(w, q) = Hii(w, q)

Because cost function is concave: A general property

Therefore:
Cii(w, q)  0
Hii(w, q)  0
The relationship of conditional
demand for an input with its own
price cannot be positive
and so…
March 2012
Frank Cowell: Firm- Demand & Supply
11
Conditional input demand curve
w1
 Consider the demand for input 1
 Consequence of result 2?
H1(w,q)
 “Downward-sloping”
conditional demand
 In some cases it is also
possible that Hii = 0
H11(w, q) < 0
z1
March 2012
 Corresponds to the
case where isoquant is
kinked: multiple w values
consistent with same z*
Frank Cowell: Firm- Demand & Supply
12
For the conditional demand function…
 Nonconvex Z yields discontinuous H
 Cross-price effects are symmetric
 Own-price demand slopes downward
 (exceptional case: own-price demand could be constant)
March 2012
Frank Cowell: Firm- Demand & Supply
13
Overview…
Firm: Comparative
Statics
Conditional
Input Demand
Response function
for stage 2
optimisation
Output
Supply
Ordinary
Input Demand
Short-run
problem
March 2012
Frank Cowell: Firm- Demand & Supply
14
The second response function

Review the profit-maximisation
problem and its solution
 Choose q to maximise:
The “stage 2” problem
pq – C (w, q)

From the FOC:
p  Cq (w, q*)
pq*  C(w, q*)

“Price equals marginal cost”
“Price covers average cost”
profit-maximising value for output:
 S is the supply function
q* = S (w, p)
input
prices
March 2012
output
price
(again it may actually be a
correspondence)
Frank Cowell: Firm- Demand & Supply
15
Supply of output and output price
 Use the FOC:
Cq (w, q) = p
“marginal cost equals price”
 Use the supply function for q:
Cq (w, S(w, p) ) = p
Gives an equation in w and p
Differential of S
with respect to p
 Differentiate with respect to p Use the “function of a function” rule
Cqq (w, S(w, p) ) Sp (w, p) = 1
Positive if MC is
increasing
 Rearrange:
1
.
Sp (w, p) = ———— Gives slope of supply function
Cqq (w, q)
March 2012
Frank Cowell: Firm- Demand & Supply
16
The firm’s supply curve
p
 The firm’s AC and MC curves
 For given p read off optimal q*
 Continue down to p
 What happens below p
Cq
C/q
 Case illustrated is
for f with first IRTS,
then DRTS
Response is a
discontinuous map:
jumps in q*
Multiple q* at
this price
_p –
no price yields
a solution here
March 2012
Supply response is
given by q=S(w,p)
|
_q
q Map is multivalued
at the discontinuity
Frank Cowell: Firm- Demand & Supply
17
Supply of output and price of input j
 Use the FOC:
Cq (w, S(w, p) ) = p
Same as before: “price
equals marginal cost”
Use the “function of a function”
 Differentiate with respect to wj
Cqj(w, q*) + Cqq (w, q*) Sj(w, p) = 0 rule again
 Rearrange:
Cqj(w, q*)
Sj(w, p) = – ————
Cqq(w, q*)
Supply of output must fall
with wj if marginal cost
increases with wj
Remember, this is
positive
March 2012
Frank Cowell: Firm- Demand & Supply
18
For the supply function…
 Supply curve slopes upward
 Supply decreases with the price of an input, if MC increases
with the price of that input
 Nonconcave f yields discontinuous S
 IRTS means f is nonconcave and so S is discontinuous
March 2012
Frank Cowell: Firm- Demand & Supply
19
Overview…
Firm: Comparative
Statics
Conditional
Input Demand
Response function for
combined
optimisation problem
Output
Supply
Ordinary
Input Demand
Short-run
problem
March 2012
Frank Cowell: Firm- Demand & Supply
20
The third response function
 Recall the first two response
functions:

zi* = Hi(w,q)
Demand for input i,
conditional on output q
q* = S (w, p)
Supply of output
Now substitute for q* :
zi* = Hi(w, S(w, p) )

Use this to define a new function:
Di(w,p) := Hi(w, S(w, p) )
input
prices
March 2012
Stages 1 & 2 combined…
output
price
Demand for input i (unconditional )
Use this relationship to
analyse further the firm’s
response to price changes
Frank Cowell: Firm- Demand & Supply
21
Demand for i and the price of output
 Take the relationship
Di(w, p) = Hi(w, S(w, p))
 Differentiate with respect to p:
Dpi(w, p) = Hqi(w, q*) Sp(w, p)
Di increases with p iff Hi
increases with q. Reason? Supply
increases with price ( Sp>0)
“function of a
function” rule again
 But we also have, for any q:
Hi(w, q) = Ci(w, q)
Hqi (w, q) = Ciq(w, q)
Shephard’s Lemma again
 Substitute in the above:
Dpi(w, p) = Cqi(w, q*)Sp(w, p)
Demand for input i (Di) increases
with p iff marginal cost (Cq)
increases with wi
March 2012
Frank Cowell: Firm- Demand & Supply
22
Demand for i and the price of j
 Again take the relationship
Di(w, p) = Hi(w, S(w, p))
 Differentiate with respect to wj:
Dji(w, p) = Hji(w, q*) + Hqi(w, q*)Sj(w, p)
 Use Shephard’s Lemma again:
Hqi(w, q) = Ciq(w, q) = Cqi(w, q)
 Use this and the previous result on Sj(w, p) to give a
decomposition into a “substitution effect” and an “output effect”:
Dji(w, p) = Hji(w, q*) 
Cjq(w, q*)
 Ciq(w, q*)
Cqq(w, q*)
“output
effect”
.
“substitution
effect”
March 2012
Frank Cowell: Firm- Demand & Supply
23
Results from decomposition formula
 Take the general relationship:
Ciq(w, q*)Cjq(w, q*)
Dji(w, p) = Hji(w, q*)  
Cqq(w, q*)
.
We already know
this is symmetric in i
and j.
The effect wi on demand for
input j equals the effect of wj
on demand for input i
Obviously
symmetric in i
and j.
 Now take the special case where j = i:
Ciq(w, q*)2
Dii(w, p) = Hii(w, q*)  
Cqq(w, q*)
.
We already know this
is negative or zero.
March 2012
If wi increases, the demand for
input i cannot rise
cannot be
positive.
Frank Cowell: Firm- Demand & Supply
24
Input-price fall: substitution effect
 The initial equilibrium
w1
 price of input falls
conditional demand
curve
original
 value to firm of price fall,
given a fixed output level
output level
H1(w,q)
price
fall
initial price
level
Change in cost
Notional increase in
factor input if output
target is held constant
z1*
March 2012
z1
Frank Cowell: Firm- Demand & Supply
25
Input-price fall: total effect
w1
Conditional
demand at
original output
 The initial equilibrium
 Substitution effect of inputprice of fall
 Total effect of input-price fall
Conditional
demand at new
output
price
fall
initial price
level
ordinary demand
curve
z1*
March 2012
z**
1
z1
Frank Cowell: Firm- Demand & Supply
26
The ordinary demand function…
 Nonconvex Z may yield a discontinuous D
 Cross-price effects are symmetric
 Own-price demand slopes downward
 Same basic properties as for H function
March 2012
Frank Cowell: Firm- Demand & Supply
27
Overview…
Firm: Comparative
Statics
Conditional
Input Demand
Optimisation subject
to side-constraint
Output
Supply
Ordinary
Input Demand
Short-run
problem
March 2012
Frank Cowell: Firm- Demand & Supply
28
The short run…
 This is not a moment in time but…
 … is defined by additional constraints within the model
 Counterparts in other economic applications where we
sometimes need to introduce side constraints
March 2012
Frank Cowell: Firm- Demand & Supply
29
The short-run problem
 We build on the firm’s standard optimisation
problem
 Choose q and z to maximise
m
P := pq – S wizi
i=1
 subject to the standard constraints:
q  f (z)
q  0, z  0
 But we add a side condition to this problem:
zm = `zm
 Let `q be the value of q for which zm =`zm would have
been freely chosen in the unrestricted cost-min
problem…
March 2012
Frank Cowell: Firm- Demand & Supply
30
The short-run cost function
~
_
 C(w, q, zm ) := min S wi zi
{zm =`zm }
 Short-run demand for input i:
~
_
~
_
i
H (w, q, zm) =Ci(w, q, zm )
 Compare with the ordinary cost
function
~
_
C(w, q)  C(w, q, zm )
 So, dividing by q:
~
_
C(w,
q) C(w,
q, zm )
_______
_________

q
q
March 2012
The solution function with
the side constraint
Follows from Shephard’s Lemma
By definition of the cost
function. We have “=” if q =`q
Short-run AC ≥ long-run AC.
SRAC = LRAC at q =`q
Supply curves
Frank Cowell: Firm- Demand & Supply
31
MC, AC and supply in the short and long
run
 AC if all inputs are variable
 MC if all inputs are variable
 Fix an output level
p
 AC if input m is now kept fixed
~
Cq
 MC if input m is now kept fixed
 Supply curve in long run
Cq
~
C/q
 Supply curve in short run
C/q
SRAC touches LRAC at
the given output

SRMC cuts LRMC at
the given output
q
March 2012
q
The supply curve is
steeper in the short run
Frank Cowell: Firm- Demand & Supply
32
Conditional input demand
 The original demand curve for input 1
w1
H1(w,q)
 The demand curve from the problem
with the side constraint
 “Downward-sloping” conditional
demand
 Conditional demand curve is
steeper in the short run
~
_
H1(w, q, zm)
z1
March 2012
Frank Cowell: Firm- Demand & Supply
33
Key concepts
 Basic functional relations
 price signals  firm  input/output responses

Hi(w,q)
demand for input i,
conditional on output
Review

S (w,p)
supply of output
Review

Di(w,p)
demand for input i
(unconditional )
Review
And they all hook together like this:
 Hi(w, S(w,p)) = Di(w,p)
March 2012
Frank Cowell: Firm- Demand & Supply
34
What next?
 Analyse the firm under a variety of market conditions
 Apply the analysis to the consumer’s optimisation problem
March 2012
Frank Cowell: Firm- Demand & Supply
35